Theorem rnffi

Description: The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	rnffi	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐴 ∈ Fin ) → ran 𝐹 ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		ffi	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐴 ∈ Fin ) → 𝐹 ∈ Fin )
2		rnfi	⊢ ( 𝐹 ∈ Fin → ran 𝐹 ∈ Fin )
3	1 2	syl	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐴 ∈ Fin ) → ran 𝐹 ∈ Fin )